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Congrats to Rieselsieve!

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  • Jean PennĂ©
    Hi, 22a 467917*2^1993429-1 600088 L160 2005 Congrats to the Rieselsieve project and all its participants ! 43th Mersenne prime found, more than 2
    Message 1 of 13 , Dec 25, 2005
      Hi,

      22a 467917*2^1993429-1 600088 L160 2005

      Congrats to the Rieselsieve project and all its participants !

      43th Mersenne prime found, more than 2 million bits 3-2-1 success...

      that was a good week for the prime number searchers!

      What would happen for the last week of 2005 year ?

      I wish an happy new year for all !

      Jean
    • ed pegg
      The next Al Zimmermann Programming contest might be about Prime Generating Polynomials. It s currently running neck and neck with Protein Folding.
      Message 2 of 13 , Jan 2, 2006
        The next Al Zimmermann Programming contest might be about
        Prime Generating Polynomials. It's currently running neck
        and neck with Protein Folding.
        http://www.recmath.com/contest/votes.php

        Under the rules I'm proposing for the contest, here are the
        current known record holders for Prime Generating Polynomials,
        orders 1 to 4. Orders 5 and up seem to be unexplored.

        1) 44546738095860 n + 56211383760397 Score:23 n=0..22 Frind
        2) 36 n^2 - 810 n + 2753 Score:45 n=0..44 Ruby
        3) 3 n^3 - 183 n^2 + 3318 n - 18757 Score:43 n=0..46 Ruiz
        4) n^4 + 29 n^2 + 101 Score:20 n=0..19 Pegg

        The scoring rules:
        1. Polynomial f(k) must produce primes from 0 to n.
        2. The score will be the number of *distinct* primes
        when |f(k)| is evaluated from 0 to n.
        3. In case of a tie, the lower value (tighter value) of n wins.
        4. In case of a tie, the product of non-zero coefficients will
        be evaluated, and the lowest product wins.

        My own result for order-4 polynomials will likely be surpassed
        very easily. If you would like to vote for this contest, please
        visit http://www.recmath.com/contest/ . The winners will split
        up $500.

        Ed Pegg Jr
      • Mark Underwood
        Here s one that won t be a winner but is fairly long. I can t find it in the Integer Sequences, so maybe it s new : 4x^2 - 212x + 2411 Generates 27 distinct
        Message 3 of 13 , Jan 3, 2006
          Here's one that won't be a winner but is fairly long. I can't find it
          in the Integer Sequences, so maybe it's 'new':

          4x^2 - 212x + 2411

          Generates 27 distinct primes from x=0 (p=2411) to x=26 (p=-397)
          before doubling back. So it's prime from x=0 to x=53. It met its end
          at the hands of a factor of 37.

          Mark



          --- In primenumbers@yahoogroups.com, ed pegg <ed@m...> wrote:
          >
          > The next Al Zimmermann Programming contest might be about
          > Prime Generating Polynomials. It's currently running neck
          > and neck with Protein Folding.
          > http://www.recmath.com/contest/votes.php
          >
          > Under the rules I'm proposing for the contest, here are the
          > current known record holders for Prime Generating Polynomials,
          > orders 1 to 4. Orders 5 and up seem to be unexplored.
          >
          > 1) 44546738095860 n + 56211383760397 Score:23 n=0..22 Frind
          > 2) 36 n^2 - 810 n + 2753 Score:45 n=0..44 Ruby
          > 3) 3 n^3 - 183 n^2 + 3318 n - 18757 Score:43 n=0..46 Ruiz
          > 4) n^4 + 29 n^2 + 101 Score:20 n=0..19 Pegg
          >
          > The scoring rules:
          > 1. Polynomial f(k) must produce primes from 0 to n.
          > 2. The score will be the number of *distinct* primes
          > when |f(k)| is evaluated from 0 to n.
          > 3. In case of a tie, the lower value (tighter value) of n wins.
          > 4. In case of a tie, the product of non-zero coefficients will
          > be evaluated, and the lowest product wins.
          >
          > My own result for order-4 polynomials will likely be surpassed
          > very easily. If you would like to vote for this contest, please
          > visit http://www.recmath.com/contest/ . The winners will split
          > up $500.
          >
          > Ed Pegg Jr
          >
        • Mark Underwood
          Here s another one: 4x^2+12x-1583 is prime from x=0 to x=34, a span of 35 unique primes increasing from -1583 to 3449. It too met its end at the hands of
          Message 4 of 13 , Jan 3, 2006
            Here's another one:

            4x^2+12x-1583

            is prime from x=0 to x=34, a span of 35 unique primes increasing
            from -1583 to 3449. It too met its end at the hands of factor 37,
            which is still at large.

            Mark



            --- In primenumbers@yahoogroups.com, "Mark Underwood"
            <mark.underwood@s...> wrote:
            >
            >
            > Here's one that won't be a winner but is fairly long. I can't find
            it
            > in the Integer Sequences, so maybe it's 'new':
            >
            > 4x^2 - 212x + 2411
            >
            > Generates 27 distinct primes from x=0 (p=2411) to x=26 (p=-397)
            > before doubling back. So it's prime from x=0 to x=53. It met its
            end
            > at the hands of a factor of 37.
            >
            > Mark
            >
            >
            >
            > --- In primenumbers@yahoogroups.com, ed pegg <ed@m...> wrote:
            > >
            > > The next Al Zimmermann Programming contest might be about
            > > Prime Generating Polynomials. It's currently running neck
            > > and neck with Protein Folding.
            > > http://www.recmath.com/contest/votes.php
            > >
            > > Under the rules I'm proposing for the contest, here are the
            > > current known record holders for Prime Generating Polynomials,
            > > orders 1 to 4. Orders 5 and up seem to be unexplored.
            > >
            > > 1) 44546738095860 n + 56211383760397 Score:23 n=0..22 Frind
            > > 2) 36 n^2 - 810 n + 2753 Score:45 n=0..44 Ruby
            > > 3) 3 n^3 - 183 n^2 + 3318 n - 18757 Score:43 n=0..46 Ruiz
            > > 4) n^4 + 29 n^2 + 101 Score:20 n=0..19 Pegg
            > >
            > > The scoring rules:
            > > 1. Polynomial f(k) must produce primes from 0 to n.
            > > 2. The score will be the number of *distinct* primes
            > > when |f(k)| is evaluated from 0 to n.
            > > 3. In case of a tie, the lower value (tighter value) of n wins.
            > > 4. In case of a tie, the product of non-zero coefficients will
            > > be evaluated, and the lowest product wins.
            > >
            > > My own result for order-4 polynomials will likely be surpassed
            > > very easily. If you would like to vote for this contest, please
            > > visit http://www.recmath.com/contest/ . The winners will split
            > > up $500.
            > >
            > > Ed Pegg Jr
            > >
            >
          • Jacques Tramu
            36 x^2 - 666 x + 1277 is prime from x = 0 to x = 42 . (different values) Note the use of the well known numbers 666 and 42 . JT 0 1277 1 647 2
            Message 5 of 13 , Jan 4, 2006
              36 x^2 - 666 x + 1277 is prime from x = 0 to x = 42 . (different values)

              Note the use of the well known numbers '666' and '42' .

              JT

              0 1277
              1 647
              2 89
              3 -397
              4 -811
              5 -1153
              6 -1423
              7 -1621
              8 -1747
              9 -1801
              10 -1783
              11 -1693
              12 -1531
              13 -1297
              14 -991
              15 -613
              16 -163
              17 359
              18 953
              19 1619
              20 2357
              21 3167
              22 4049
              23 5003
              24 6029
              25 7127
              26 8297
              27 9539
              28 10853
              29 12239
              30 13697
              31 15227
              32 16829
              33 18503
              34 20249
              35 22067
              36 23957
              37 25919
              38 27953
              39 30059
              40 32237
              41 34487
              42 36809
              43 197*199


              ------------------------------------
              http://www.echolalie.com
            • Mark Underwood
              ... values) ... What a beastly formula, Jacques! It is actually prime from x=-2 to 42, making 45 distinct prime values! For x=0 to x=44 your formula would
              Message 6 of 13 , Jan 6, 2006
                --- In primenumbers@yahoogroups.com, "Jacques Tramu"
                <jacques.tramu@e...> wrote:
                >
                > 36 x^2 - 666 x + 1277 is prime from x = 0 to x = 42 . (different
                values)
                >
                > Note the use of the well known numbers '666' and '42' .
                >(snip)
                > JT


                What a 'beastly' formula, Jacques! It is actually prime from x=-2 to
                42, making 45 distinct prime values! For x=0 to x=44 your formula
                would become

                36x^2 - 810 + 2753

                which would not make the devil happy ;) Incredibly your expression
                never yields prime factors below 59.

                Mark
              • Jacques Tramu
                Thx Mark. Halas, after checking http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html I found, and you can see that 36x^2 - 810 + 2753 is an already
                Message 7 of 13 , Jan 6, 2006
                  Thx Mark.

                  Halas, after checking
                  http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html

                  I found, and you can see that
                  36x^2 - 810 + 2753 is an already known record (Fung and Ruby)

                  (Sloane sequence : A060268)

                  So, it appears that I discovered the '666' version of it , without knowing
                  A060268.
                  Sorry,
                  Jacques.



                  ----- Original Message -----
                  From: "Mark Underwood" <mark.underwood@...>
                  "
                  > <jacques.tramu@e...> wrote:
                  >>
                  >> 36 x^2 - 666 x + 1277 is prime from x = 0 to x = 42 . (different
                  > values)
                  >>
                  > What a 'beastly' formula, Jacques! It is actually prime from x=-2 to
                  > 42, making 45 distinct prime values! For x=0 to x=44 your formula
                  > would become
                  >
                  > 36x^2 - 810 + 2753
                  >
                  > which would not make the devil happy ;) Incredibly your expression
                  > never yields prime factors below 59.
                  >
                  > Mark
                  >
                  ------------------------------------
                  http://www.echolalie.com
                • Dick
                  ... A quadratic expression that yields no prime factors less than a given value is easy to generate arbitrarily. Start with any integer valued f(x)=ax^2+bx+c
                  Message 8 of 13 , Jan 6, 2006
                    --- In primenumbers@yahoogroups.com, "Mark Underwood"
                    <mark.underwood@s...> wrote:

                    > which would not make the devil happy ;) Incredibly your expression
                    > never yields prime factors below 59.


                    A quadratic expression that yields no prime factors less than a given
                    value is easy to generate arbitrarily.

                    Start with any integer valued f(x)=ax^2+bx+c and divide through by any
                    common factor. Let's say f(d) is odd and f(d+1) is even, then take
                    the three seed values f(d), f(d+2),f(d+4) and solve for the
                    coefficients of the new quadratic, f'(x) (you have 3 eqs, 3 unknowns,
                    so this can always be done).

                    If 3 appears as a factor of f'(x), only 2 of 3 consecutive solutions
                    of f'(x) can possibly be divisible by 3. Choose one that is not, say
                    it is f'(e), then take 3 consecutive values f'(e),f'(e+3),f'(e+6) and
                    solve for the quadratic f''(x). Continue as high as you please. Any
                    prime factor q, will appear at most twice within q consecutive values
                    of the quadratic progression, so there are always at least q-2
                    "paralell" quadratic progressions where q cannot be a divisor of the
                    quadratic.

                    -Dick Boland
                  • Dick
                    ... Here s another approach that yields the same result more directly, but causes the quadratic to grow large more quickly. Choose your threshold prime, say
                    Message 9 of 13 , Jan 6, 2006
                      --- In primenumbers@yahoogroups.com, "Dick" <richard042@y...> wrote:
                      > A quadratic expression that yields no prime factors less than a given
                      > value is easy to generate arbitrarily.


                      Here's another approach that yields the same result more directly, but
                      causes the quadratic to grow large more quickly. Choose your
                      threshold prime, say p(a), then take any three integers, t,u,v where
                      each is co-prime to all primes<p(a+1), for example t=p(a+1), u=p(a+2),
                      v=p(a+3) then solve for the coefficients of quadratic f(x) given by 3
                      seed values f(0)=t, f(1)=u, f(2)=v. Now solve for the quadratic f'(x)
                      given by 3 seed values f(a#),f(2a#),f(3a#). f'(x) will then have no
                      prime divisors <p(a+1).

                      -Dick Boland
                    • Dick
                      ... Of course, it would only be necessary that t have no prime divisors
                      Message 10 of 13 , Jan 6, 2006
                        --- In primenumbers@yahoogroups.com, "Dick" <richard042@y...> wrote:
                        > --- In primenumbers@yahoogroups.com, "Dick" <richard042@y...> wrote:

                        > threshold prime, say p(a), then take any three integers, t,u,v where
                        > each is co-prime to all primes<p(a+1), for example t=p(a+1), u=p(a+2),
                        > v=p(a+3) then solve for the coefficients of quadratic f(x) given by 3
                        > seed values f(0)=t, f(1)=u, f(2)=v. Now solve for the quadratic f'(x)
                        > given by 3 seed values f(a#),f(2a#),f(3a#). f'(x) will then have no
                        > prime divisors <p(a+1).


                        Of course, it would only be necessary that t have no prime
                        divisors<p(a+1) in order for f'(x) to also have this property.

                        -Dick Boland
                      • Mark Underwood
                        ... No probs Jacques. I did a retake of Ed Pegg s post which originated this thread, and there was Ruby s expression! But a guru has just informed me that
                        Message 11 of 13 , Jan 6, 2006
                          --- In primenumbers@yahoogroups.com, "Jacques Tramu"
                          <jacques.tramu@e...> wrote:
                          >
                          > Thx Mark.
                          >
                          > Halas, after checking
                          > http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html
                          >
                          > I found, and you can see that
                          > 36x^2 - 810 + 2753 is an already known record (Fung and Ruby)
                          >
                          > (Sloane sequence : A060268)

                          No probs Jacques. I did a retake of Ed Pegg's post which originated
                          this thread, and there was Ruby's expression!

                          But a guru has just informed me that Ruby's expression is no longer a
                          record - it now might be held by Dror Speiser who has 46 different
                          primes in the expression

                          103x^2 - 4707x + 50383

                          from x=0 to x=45. (!) See
                          http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0506&L=NMBRTHRY&P=R260

                          This expression as well as Ruby's generates no prime factors below 59.

                          I just stumbled across one that has no prime factors below 61 :
                          x^2 - 479x +3851
                          but it generates only 21 consecutive primes.

                          Mark
                        • Mark Underwood
                          It appears there has been a retraction and the expression 103x^2 - 4707x + 50383 is prime for only 43 primes. It is apparently tied for second place to
                          Message 12 of 13 , Jan 6, 2006
                            It appears there has been a retraction and the expression
                            103x^2 - 4707x + 50383
                            is prime for only 43 primes. It is apparently tied for second
                            place to
                            47x^2-1701x+10181,
                            yet another one discovered by Ruby and Fung.

                            It's nice to see that I'm not the only one who can look at data and
                            miss what I don't want to see (ie, a composite). ;)


                            Mark



                            --- In primenumbers@yahoogroups.com, "Mark Underwood"
                            <mark.underwood@s...> wrote:
                            >
                            > --- In primenumbers@yahoogroups.com, "Jacques Tramu"
                            > <jacques.tramu@e...> wrote:
                            > >
                            > > Thx Mark.
                            > >
                            > > Halas, after checking
                            > > http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html
                            > >
                            > > I found, and you can see that
                            > > 36x^2 - 810 + 2753 is an already known record (Fung and Ruby)
                            > >
                            > > (Sloane sequence : A060268)
                            >
                            > No probs Jacques. I did a retake of Ed Pegg's post which originated
                            > this thread, and there was Ruby's expression!
                            >
                            > But a guru has just informed me that Ruby's expression is no longer
                            a
                            > record - it now might be held by Dror Speiser who has 46 different
                            > primes in the expression
                            >
                            > 103x^2 - 4707x + 50383
                            >
                            > from x=0 to x=45. (!) See
                            > http://listserv.nodak.edu/cgi-bin/wa.exe?
                            A2=ind0506&L=NMBRTHRY&P=R260
                            >
                            > This expression as well as Ruby's generates no prime factors below
                            59.
                            >
                            > I just stumbled across one that has no prime factors below 61 :
                            > x^2 - 479x +3851
                            > but it generates only 21 consecutive primes.
                            >
                            > Mark
                            >
                          • Jens Kruse Andersen
                            ... Yes. Here is PARI/GP code: f(n)=local(c,p,i,r);c=1;r=2;forprime(p=3,n,i=1; while(issquare(Mod(i,p)),i++);c+=lift((Mod(1-i,p)/4-c)/r)*r;r*=p);c x^2+x+f(n)
                            Message 13 of 13 , Jan 6, 2006
                              Dick wrote:

                              > A quadratic expression that yields no prime factors less than a given
                              > value is easy to generate arbitrarily.

                              Yes. Here is PARI/GP code:

                              f(n)=local(c,p,i,r);c=1;r=2;forprime(p=3,n,i=1;
                              while(issquare(Mod(i,p)),i++);c+=lift((Mod(1-i,p)/4-c)/r)*r;r*=p);c

                              x^2+x+f(n) never has a factor <= n.
                              f(n) is a little below n# (which is near e^n).
                              f(277) has 113 digits:
                              52211040781253690937101509813868236062339335365181118768\
                              264647548098518281973426206257327260311399656484832821211

                              f(100000) is computed in 2 seconds and has 43293 digits.
                              It is far harder to find the smallest constants to avoid
                              all small factors in x^2+x+A.
                              http://www.primepuzzles.net/conjectures/conj_017.htm lists that.
                              The record is A=2457080965043150051 which gives 281 as smallest factor.

                              --
                              Jens Kruse Andersen
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